The following airlines are operating flights to Madeira Island:

Lufthansa | Flights from all major airports in Germany/ Austria and Switzerland.
TAP Portugal | TAP Portugal offers good scheduled flight connections from all major European cities. For instance, TAP flies from 6 big German cities (Berlin, Düsseldorf, Hamburg, Hannover, Frankfurt and Munich), as well as Vienna/ Austria and Switzerland (Zurich, Geneva) via Lissabon to Funchal and Porto Santo, for the most part several times a day.
Easyjet | There are flights coming from London, Lisbon and Bristol.
Finnair | Many flights operating from the Scandic region to Funchal.
Transavia | Flights between Amsterdam and Funchal
Monarch | Flights between UK and Funchal